Modified large deviation figure.

6 files changed, 1681 insertions, 1581 deletions

diff --git a/figs/large_deviation.pdf b/figs/large_deviation.pdf
index 9149a5a..29424c2 100644
--- a/figs/large_deviation.pdf
+++ b/figs/large_deviation.pdf
diff --git a/figs/spectrum_eq.pdf b/figs/spectrum_eq.pdf
new file mode 100644
index 0000000..69db15c
--- /dev/null
+++ b/figs/spectrum_eq.pdf
diff --git a/figs/spectrum_less.pdf b/figs/spectrum_less.pdf
new file mode 100644
index 0000000..439d485
--- /dev/null
+++ b/figs/spectrum_less.pdf
diff --git a/figs/spectrum_more.pdf b/figs/spectrum_more.pdf
new file mode 100644
index 0000000..8db227a
--- /dev/null
+++ b/figs/spectrum_more.pdf
diff --git a/figures.nb b/figures.nb
index 411cf9a..3a5b708 100644
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-           "PlotRange" -> {{-15., 3.}, {0., 0.12732395447351627`}}, 
-           "Frame" -> {{False, False}, {False, False}}, 
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-           "Axes" -> {True, True}, "LabelStyle" -> {FontFamily -> "Helvetica", 
-             GrayLevel[0], FontSize -> 12}, "AspectRatio" -> Rational[2, 5], 
+           "PlotRange" -> {{-3., 15.}, {0., 0.16976527263135505`}}, 
+           "Frame" -> {{True, True}, {True, True}}, "AxesOrigin" -> {0, 0}, 
+           "ImageSize" -> {118, 
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+           "LabelStyle" -> {FontFamily -> "Times", 
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-           "PlotRange" -> {{-15., 3.}, {0., 0.12732395447351627`}}, 
-           "Frame" -> {{False, False}, {False, False}}, 
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-           "Axes" -> {True, True}, "LabelStyle" -> {FontFamily -> "Helvetica", 
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+           "PlotRange" -> {{-3., 15.}, {0., 0.16976527263135505`}}, 
+           "Frame" -> {{True, True}, {True, True}}, "AxesOrigin" -> {0, 0}, 
+           "ImageSize" -> {118, 
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+           "LabelStyle" -> {FontFamily -> "Times", 
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diff --git a/marginal.tex b/marginal.tex
index c461f81..e2bd24f 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -288,13 +288,27 @@ corresponds to the intersection of the average spectrum with zero, i.e., a
 pseudogap.
 
 \begin{figure}
-  \includegraphics[width=\columnwidth]{figs/large_deviation.pdf}
+  \hspace{1.3em}
+  \includegraphics{figs/spectrum_less.pdf}
+  \hspace{-2em}
+  \includegraphics{figs/spectrum_eq.pdf}
+  \hspace{-2em}
+  \includegraphics{figs/spectrum_more.pdf}
+  \\
+  \includegraphics{figs/large_deviation.pdf}
   \caption{
-    The large deviation function $G_\sigma(\mu)$ defined in
+    The large deviation function $G_0(\mu)$ defined in
     \eqref{eq:large.dev} as a function of the shift $\mu$ to the
-    GOE diagonal. As expected, $G_\sigma(2\sigma)=0$, while for
+    GOE diagonal. $G_0(2\sigma)=0$, while for
     $\mu>2\sigma$ it is negative and for $\mu<2\sigma$ it gains an
-    imaginary part.
+    imaginary part. The top panels show schematically what happens to the
+    spectral density in each of these regimes. For $\mu<2\sigma$, an $N^2$
+    large deviation would be required to fix the smallest eigenvalue to zero
+    and the calculation breaks down, leading to the imaginary part. For
+    $\mu>2\sigma$ the spectrum can satisfy the constraint on the smallest
+    eigenvalue by isolating a single eigenvalue at zero at the cost of an
+    order-$N$ large deviation. At $\mu=2\sigma$, the transition point, the
+    spectrum is pseudogapped or marginal.
   } \label{fig:large.dev}
 \end{figure}